Sharply o-minimal structures and sharp cellular decomposition
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866910030404517888 |
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| author | Binyamini, Gal Novikov, Dmitri Zak, Benny |
| author_facet | Binyamini, Gal Novikov, Dmitri Zak, Benny |
| contents | Sharply o-minimal structures (denoted \so-minimal) are a strict subclass of the o-minimal structures, aimed at capturing some finer features of structures arising from algebraic geometry and Hodge theory. Sharp o-minimality associates to each definable set a pair of integers known as \emph{format} and \emph{degree}, similar to the ambient dimension and degree in the algebraic case; gives bounds on the growth of these quantities under the logical operations; and allows one to control the geometric complexity of a set in terms of its format and degree. These axioms have significant implications on arithmetic properties of definable sets -- for example, \so-minimality was recently used by the authors to settle Wilkie's conjecture on rational points in $\mathbb{R}_{\exp}$-definable sets.
In this paper we develop some basic theory of sharply o-minimal structures. We introduce the notions of reduction and equivalence on the class of \so-minimal structures. We give three variants of the definition of \so-minimality, of increasing strength, and show that they all agree up to reduction. We also consider the problem of ``sharp cell decomposition'', i.e. cell decomposition with good control on the number of the cells and their formats and degrees. We show that every \so-minimal structure can be reduced to one admitting sharp cell decomposition, and use this to prove bounds on the Betti numbers of definable sets in terms of format and degree. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_10972 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Sharply o-minimal structures and sharp cellular decomposition Binyamini, Gal Novikov, Dmitri Zak, Benny Logic Algebraic Geometry 14Gxx, 11Jxx, 03Cxx Sharply o-minimal structures (denoted \so-minimal) are a strict subclass of the o-minimal structures, aimed at capturing some finer features of structures arising from algebraic geometry and Hodge theory. Sharp o-minimality associates to each definable set a pair of integers known as \emph{format} and \emph{degree}, similar to the ambient dimension and degree in the algebraic case; gives bounds on the growth of these quantities under the logical operations; and allows one to control the geometric complexity of a set in terms of its format and degree. These axioms have significant implications on arithmetic properties of definable sets -- for example, \so-minimality was recently used by the authors to settle Wilkie's conjecture on rational points in $\mathbb{R}_{\exp}$-definable sets. In this paper we develop some basic theory of sharply o-minimal structures. We introduce the notions of reduction and equivalence on the class of \so-minimal structures. We give three variants of the definition of \so-minimality, of increasing strength, and show that they all agree up to reduction. We also consider the problem of ``sharp cell decomposition'', i.e. cell decomposition with good control on the number of the cells and their formats and degrees. We show that every \so-minimal structure can be reduced to one admitting sharp cell decomposition, and use this to prove bounds on the Betti numbers of definable sets in terms of format and degree. |
| title | Sharply o-minimal structures and sharp cellular decomposition |
| topic | Logic Algebraic Geometry 14Gxx, 11Jxx, 03Cxx |
| url | https://arxiv.org/abs/2209.10972 |